Representing Piecewise-Linear Functions by Functions with Minimal Arity
This work provides a theoretical foundation for understanding the complexity of representing piecewise-linear functions, which is incremental as it extends prior results on tight bounds.
The paper establishes a correspondence between a continuous piecewise-linear function and the minimal number of arguments needed in its decomposition as a linear combination of max functions, showing that this number is directly connected to the tessellation of the input space induced by the function.
Any continuous piecewise-linear function $F\colon \mathbb{R}^{n}\to \mathbb{R}$ can be represented as a linear combination of $\max$ functions of at most $n+1$ affine-linear functions. In our previous paper [``Representing piecewise linear functions by functions with small arity'', AAECC, 2023], we showed that this upper bound of $n+1$ arguments is tight. In the present paper, we extend this result by establishing a correspondence between the function $F$ and the minimal number of arguments that are needed in any such decomposition. We show that the tessellation of the input space $\mathbb{R}^{n}$ induced by the function $F$ has a direct connection to the number of arguments in the $\max$ functions.